Chronological transitions of hepatocyte growth factor treatment effects in spinal cord injury tissue

Inflammatory responses are known to suppress neural regeneration in patients receiving stem cell-based regenerative therapy for spinal cord injury (SCI). Consequently, pathways involved in neurogenesis and immunomodulation, such as the hepatocyte growth factor (HGF)/MET signaling cascade, have garnered significant attention. Notably, various studies, including our own, have highlighted the enhanced recovery of locomotor functions achieved in SCI animal models by combining HGF pretreatment and human induced stem cell-derived neural stem/progenitor cell (hiPSC-NS/PC) transplantation. However, these studies implicitly hypothesized that the functionality of HGF in SCI would be time consistent and did not elucidate its dynamics. In the present article, we investigated the time-course of the effect of HGF on SCI, aiming to uncover a more precise mechanism for HGF administration, which is indispensable for developing crystallizing protocols for combination therapy. To this end, we performed a detailed investigation of the temporal variation of HGF using the RNA-seq data we obtained in our most recent study. Leveraging the time-series design of the data, which we did not fully exploit previously, we identified three components in the effects of HGF that operate at different times: early effects, continuous effects, and delayed effects. Our findings suggested a concept where the three components together contribute to the acceleration of neurogenesis and immunomodulation, which reinforce the legitimacy of empirically fine-tuned protocols for HGF administration and advocate the novel possibility that the time-inconsistent effects of HGF progressively augment the efficacy of combined therapy. Supplementary Information The online version contains supplementary material available at 10.1186/s41232-024-00322-9.


Analogical explanation of our analysis design using vectors
To provide intuitive insights into our analysis design and to demonstrate why we conducted the comparison between the timewise trajectories of control and HGF+ samples (despite our primary focus on the temporal variation of the effect of HGF), we formulated the following hypotheses: l RNA-seq data are a matrix composed of vectors from vector space  ≔ ℝ !, where the  genes form the basis vectors ( ∈ ℕ, ℝ is the set of real numbers, and ℕ is the set of natural numbers).We applied this statement because it has been widely used in recent biological studies.PCA is a prominent example of an operation that requires data to constitute a vector space; it performs eigenvalue decomposition of the covariance matrix of the data according to the algebraic structures of Euclidean space (a real finite-dimensional vector space with the Euclidean norm) 1,2 .l The experimental factors can be denoted as vectors in , allowing for linear additivity.
Design of experiments is a statistical methodology that involves analyzing multiple factors, and various experimental designs, such as factorial designs and orthogonal arrays (Taguchi designs), have been developed to ensure the orthogonality of experimental factors-meaning that they exhibit no correlations and are linearly additive 3,4 .Suematsu et al. employed a 2x2 factorial design (also known as the L4 orthogonal array) for the RNA-seq data.Therefore, we can assume that the main effects of the factors (the time course and HGF administration) are orthogonal.
l Vector orientations and magnitudes intuitively correspond to either qualitative or quantitative aspects of RNA-seq data.This interpretation is inspired by GO terms, which convey abstract directions of biological functions with qualitative information (gene symbols) on collective genes.Qualitative information is a regular format for RNA-seq data analysis outputs (e.g., differentially expressed genes).Hence, this hypothesis allows us to analogize qualitative information with orientation in the realm of vector space .
l The state where two samples are biologically similar is equivalent to the state where their corresponding vectors are collinear (i.e., a scalar  ∈ ℝ such that  =  exists for vectors  and ).This assumption allows us to conceptualize the analysis for validating biological similarities using the analogy of vectors.As we hypothesized that the qualitative states of RNA-seq samples can be represented as orientations of vectors, we prefer to define semantical similarities as the parallelism of vectors (more precisely,  =  with a positive scalar  for corresponding vectors  and  rather than  = ) because the numerical aspects (magnitudes of the vectors) vanish during the qualitative assessment of RNA-seq data.Again, we emphasize that the aim of these assumptions is to provide clear-cut explanations for our analysis goals rather than to introduce mathematical notation itself.
Under the assumption of those properties, we denote the vectors in  corresponding to the respective experimental conditions (while excluding stochastic errors for simplicity).
Denoting the Day 2-control samples as  ∈  and the effect of the time course on the control samples as  ∈ , the Day 7-control samples can be denoted as  + .Moreover, defining the vector function : ℕ →  to denote the effect of HGF on an arbitrary day  ∈ ℕ as (), the Day 2-HGF+ samples was denoted as  +  (2).Considering the effect of the time course on the HGF+ samples as  * ∈ , the Day 7-HGF+ samples can be denoted as  + (2) +  * and  +  + (7).Given that, the relationship between  (2)   and (7) can be denoted as follows: In this study, our objectives were to validate 1) whether the control/HGF+ samples underwent similar transcriptomic transitions regardless of HGF administration and 2) whether the effect of HGF changed over time.Hence, we decided to quantify the intersections of upregulated (or downregulated) genes to validate the similarities between 1) the longitudinal comparisons ("Day 2-control vs. Day 7-control" and "Day 2-HGF+ vs. Day 7-HGF+") and 2) the cross-sectional comparisons ("Day 2-control vs. Day 2-HGF+" and "Day 7-control vs. Day 7-HGF+").Given all the initial assumptions, we can analogize the two objectives as the collinearity of  to  * and that of (2) to (7).With positive scalars ∃,  ∈ ℝ #$ , the following equations hold: () = (),
To illustrate the advantage of incorporating Eq. 2 into Eq. 1, let us consider a scenario where  and  * are entirely different vectors, rendering Eq. 2 inapplicable.While our primary focus is to validate the time variation in the effect of HGF ((7) − (2)), Eq. 1 becomes unsolvable under this assumption.Consequently, (7) − (2) and  * −  become indistinguishable.Even if Eq. 3 (which refers to the time consistency of the effect of HGF) holds, the presence of  * −  persists in the equation.Given that  * −  can be influenced by the effect of HGF from Day 2 through Day 7, the discussion on (7) −  (2)   becomes intricate, entangled in circular arguments involving the effect of HGF and  * − ; the discourse on (7) − (2) relies on  * − , which, in turn, depends on the effect of HGF, creating mutual dependence.

(Eq. 4)
Notably, Eq. 4 indicates that (7) − (2) can be inferred to align with the canonical timewise transition of SCI samples ( ∵ (7) − (2) ∝  ∝  * ), irrespective of other conditions, such as Eq. 3. Since the justification of Eq. 2 can be supported by the data interpretation, validating Eq. 2 provides structural simplicity to the arguments on the temporal variation of the effect of HGF.
In conclusion, we decided to compare the timewise trajectories of control and HGF+ samples prior to elucidating the temporal variation in the effect of HGF.
Anticipating that the similarity in the temporal transitions of the two groups would be a pivotal factor influencing the conceptual complexity of this study, we provided a comprehensive explanation by drawing a parallel with vector spaces.(C) Analogical explanations of our analysis design using a vector space as a coordinate system of gene expressions.Supposing the experimental conditions (time and addition of HGF) affect separately to each other so that we can decompose their effects as vectors (such as time effects or effects of HGF), our interests in this paper can be rephrased as 1) checking if the effect of time course on control samples () and that of HGF+ samples ( * ) are similar and 2) evaluating if the effect of HGF on day2 ((2)) and that of day7 ((7)) are similar.As RNA-seq analyses frequently visualize biological similarity with semantical representations (e.g., gene names, or GO terms) rather than numerical data, we consider that those schemes can be represented as

Figure S1 .
Figure S1.RNA-seq sample details and the concept of temporal variation